Subgroups — #397×D₁₀₁ #27174355

Order 12928.

820 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	2	yes	maximal
3	C₂×C₂²⋊C₄×D₁₀₁	C₂	1	1	yes	maximal, characteristic
4	#20265106	C₂	1	2	yes	maximal, complemented
5	#20265105	C₂	1	2	yes	maximal, complemented
6	#20265104	C₂	1	1	yes	maximal, characteristic, complemented
7	#20265103	C₂	1	1	yes	maximal, characteristic, complemented
8	#20265102	C₂	1	1	yes	maximal, characteristic, complemented
9	#95×D₁₀₁	C₂	1	2	yes	maximal, complemented
10	#95×D₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#397×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂⁴×D₁₀₁	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
14	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₀₁	C₂²	1	2	yes	—
16	C₂²×C₄×D₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic, complemented
18	C₂×#13108053	C₂²	1	2	yes	complemented
19	C₄⋊C₄×D₁₀₁	—	2	4	no	—
20	C₂×C₂²⋊Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×C₂²⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
22	C₂×#13108051	C₂²	1	1	yes	characteristic
23	C₂×#13108051	C₂²	1	2	yes	—
24	C₂×#13108051	C₂²	1	2	yes	—
25	C₂×#13108051	C₂²	1	1	yes	characteristic
26	C₂²⋊C₄×D₁₀₁	—	2	8	no	—
27	C₂²⋊C₄×D₁₀₁	—	2	4	no	—
28	#15692162	C₂²	1	1	yes	characteristic, complemented
29	#15692162	C₂²	1	2	yes	complemented
30	#15692162	C₂²	1	2	yes	complemented
31	#15692162	C₂²	1	2	yes	complemented
32	#15692162	C₂²	1	2	yes	complemented
33	C₂×C₄⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₁₀₁	C₂²	1	2	yes	—
36	#95×C₁₀₁	C₂²	1	1	yes	characteristic, complemented
37	#95×C₁₀₁	C₂²	1	2	yes	complemented
38	C₂³×D₁₀₁	—	2	2	no	—
39	C₂³×D₁₀₁	—	2	2	no	—
40	C₂³×D₁₀₁	—	2	4	no	—
41	C₂³×D₁₀₁	—	2	4	no	—
42	C₂³×D₁₀₁	D₄	1	4	yes	—
43	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
44	C₂³×D₁₀₁	C₂³	1	1	yes	characteristic
45	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
46	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
47	C₂²×Dic₁₀₁	C₂³	1	2	yes	—
48	C₂²×Dic₁₀₁	C₂³	1	1	yes	characteristic
49	C₂×C₄×D₁₀₁	—	2	4	no	—
50	C₂×C₄×D₁₀₁	—	2	4	no	—
51	C₂×C₄×D₁₀₁	—	2	4	no	—
52	C₂×C₄×D₁₀₁	—	2	2	no	—
53	C₂×C₄×D₁₀₁	—	2	4	no	—
54	C₂×C₄×D₁₀₁	—	2	4	no	—
55	C₂×C₄×D₁₀₁	—	2	4	no	—
56	C₂×C₄×D₁₀₁	—	2	4	no	—
57	C₂×C₄×D₁₀₁	—	2	4	no	—
58	C₂×C₄×D₁₀₁	D₄	1	4	yes	—
59	C₄⋊Dic₁₀₁	—	2	2	no	—
60	#13108053	—	2	4	no	—
61	C₂²⋊Dic₁₀₁	—	2	4	no	—
62	C₂²⋊Dic₁₀₁	—	2	2	no	—
63	#13108051	—	2	4	no	—
64	#13108051	—	2	4	no	—
65	#13108051	—	2	2	no	—
66	#13108051	—	2	2	no	—
67	C₂⁴×C₁₀₁	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
69	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
70	C₂²×C₄×C₁₀₁	C₂³	1	2	yes	—
71	C₂²×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₁₀₁	—	2	2	no	—
73	C₂²⋊C₄×C₁₀₁	—	2	4	no	—
74	C₂²⋊C₄×C₁₀₁	—	2	2	no	—
75	C₂³×C₁₀₁	—	2	1	no	—
76	C₂³×C₁₀₁	—	2	1	no	—
77	C₂³×C₁₀₁	—	2	2	no	—
78	C₂³×C₁₀₁	—	2	2	no	—
79	C₂³×C₁₀₁	C₂×D₄	1	2	yes	—
80	C₂³×C₁₀₁	C₂⁴	1	1	yes	characteristic
81	C₂²×D₁₀₁	—	2	1	no	—
82	C₂²×D₁₀₁	—	2	1	no	—
83	C₂²×D₁₀₁	—	2	2	no	—
84	C₂²×D₁₀₁	—	2	2	no	—
85	C₂²×D₁₀₁	—	2	8	no	—
86	C₂²×D₁₀₁	—	2	16	no	—
87	C₂²×D₁₀₁	—	4	4	no	—
88	C₂²×D₁₀₁	—	4	4	no	—
89	C₂²×D₁₀₁	—	4	8	no	—
90	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
91	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
92	C₂²×D₁₀₁	C₂×D₄	1	2	yes	—
93	C₂²×D₁₀₁	#41	1	2	yes	—
94	C₂²×D₁₀₁	#41	1	4	yes	—
95	C₂²×D₁₀₁	#41	1	4	yes	—
96	C₂×C₄×C₁₀₁	—	2	2	no	—
97	C₂×C₄×C₁₀₁	—	2	2	no	—
98	C₂×C₄×C₁₀₁	—	2	2	no	—
99	C₂×C₄×C₁₀₁	—	2	1	no	—
100	C₂×C₄×C₁₀₁	—	2	2	no	—
101	C₂×C₄×C₁₀₁	—	2	2	no	—
102	C₂×C₄×C₁₀₁	—	2	2	no	—
103	C₂×C₄×C₁₀₁	—	2	2	no	—
104	C₂×C₄×C₁₀₁	—	2	2	no	—
105	C₂×C₄×C₁₀₁	C₂×D₄	1	2	yes	—
106	C₂×Dic₁₀₁	—	2	2	no	—
107	C₂×Dic₁₀₁	—	2	2	no	—
108	C₂×Dic₁₀₁	—	2	2	no	—
109	C₂×Dic₁₀₁	—	2	1	no	—
110	C₂×Dic₁₀₁	—	2	2	no	—
111	C₂×Dic₁₀₁	—	2	2	no	—
112	C₂×Dic₁₀₁	—	2	2	no	—
113	C₂×Dic₁₀₁	—	2	2	no	—
114	C₂×Dic₁₀₁	—	2	2	no	—
115	C₂×Dic₁₀₁	C₂×D₄	1	2	yes	—
116	C₄×D₁₀₁	—	2	8	no	—
117	C₄×D₁₀₁	—	4	8	no	—
118	C₄×D₁₀₁	—	4	8	no	—
119	C₄×D₁₀₁	—	4	4	no	—
120	C₂²×C₁₀₁	—	2	2	no	—
121	C₂²×C₁₀₁	—	2	4	no	—
122	C₂²×C₁₀₁	—	4	1	no	—
123	C₂²×C₁₀₁	—	4	1	no	—
124	C₂²×C₁₀₁	—	4	2	no	—
125	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
126	C₂²×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
127	C₂²×C₁₀₁	C₂×#41	1	1	yes	characteristic
128	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
129	C₂²×C₁₀₁	C₂×#41	1	2	yes	—
130	C₂×D₁₀₁	—	2	2	no	—
131	C₂×D₁₀₁	—	2	4	no	—
132	C₂×D₁₀₁	—	4	1	no	—
133	C₂×D₁₀₁	—	4	1	no	—
134	C₂×D₁₀₁	—	4	2	no	—
135	C₂×D₁₀₁	—	4	16	no	—
136	C₂×D₁₀₁	#121	1	4	yes	—
137	C₂×D₁₀₁	#123	1	4	yes	—
138	C₂×D₁₀₁	#123	1	8	yes	—
139	C₂×D₁₀₁	#124	1	4	yes	—
140	C₂×D₁₀₁	#126	1	8	yes	—
141	C₄×C₁₀₁	—	2	2	no	—
142	C₄×C₁₀₁	—	4	2	no	—
143	C₄×C₁₀₁	—	4	2	no	—
144	C₄×C₁₀₁	—	4	1	no	—
145	Dic₁₀₁	—	2	2	no	—
146	Dic₁₀₁	—	4	2	no	—
147	Dic₁₀₁	—	4	2	no	—
148	Dic₁₀₁	—	4	1	no	—
149	C₂×C₁₀₁	—	4	2	no	—
150	C₂×C₁₀₁	C₂×#121	1	1	yes	characteristic
151	C₂×C₁₀₁	C₂×#123	1	1	yes	characteristic
152	C₂×C₁₀₁	C₂×#123	1	2	yes	—
153	C₂×C₁₀₁	C₂×#124	1	1	yes	characteristic
154	C₂×C₁₀₁	C₂×#126	1	2	yes	—
155	D₁₀₁	—	4	2	no	—
156	D₁₀₁	#397	1	8	yes	direct product factor
157	C₂×#397	—	101	1	no	maximal, Sylow
158	C₁₀₁	C₂×#397	1	1	yes	characteristic, complemented, Sylow
159	C₂²×C₄⋊C₄	—	101	1	no	—
160	C₂²×C₂²⋊C₄	—	101	1	no	—
161	C₂²×C₂²⋊C₄	—	101	2	no	—
162	#397	—	101	1	no	—
163	#397	—	101	1	no	—
164	#397	—	101	2	no	—
165	#397	—	101	1	no	—
166	#397	—	101	2	no	—
167	#397	D₁₀₁	1	1	yes	characteristic, direct product factor
168	C₂×#95	—	101	1	no	—
169	C₂×#95	—	101	2	no	—
170	C₂⁵	—	101	1	no	—
171	C₂³×C₄	—	101	1	no	—
172	C₂³×C₄	—	101	2	no	—
173	C₂³×C₄	—	101	1	no	—
174	C₂³×C₄	—	101	2	no	—
175	C₂×C₄⋊C₄	—	101	2	no	—
176	C₂×C₄⋊C₄	—	101	1	no	—
177	C₂×C₄⋊C₄	—	202	4	no	—
178	C₂×C₄⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic, complemented
179	C₂×C₂²⋊C₄	—	101	1	no	—
180	C₂×C₂²⋊C₄	—	101	2	no	—
181	C₂×C₂²⋊C₄	—	101	1	no	—
182	C₂×C₂²⋊C₄	—	101	2	no	—
183	C₂×C₂²⋊C₄	—	101	2	no	—
184	C₂×C₂²⋊C₄	—	101	1	no	—
185	C₂×C₂²⋊C₄	—	202	4	no	—
186	C₂×C₂²⋊C₄	—	202	8	no	—
187	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	2	yes	—
188	C₂×C₂²⋊C₄	C₂×D₁₀₁	1	1	yes	characteristic
189	#95	—	101	2	no	—
190	#95	—	101	2	no	—
191	#95	—	101	1	no	—
192	#95	—	101	2	no	—
193	#95	—	101	2	no	—
194	#95	C₂×D₁₀₁	1	2	yes	complemented
195	#95	C₂×D₁₀₁	1	1	yes	characteristic, complemented
196	C₂⁴	—	101	1	no	—
197	C₂⁴	—	101	4	no	—
198	C₂⁴	—	101	1	no	—
199	C₂⁴	—	202	2	no	—
200	C₂⁴	—	202	2	no	—
201	C₂⁴	—	202	4	no	—
202	C₂⁴	—	202	4	no	—
203	C₂⁴	C₂²×D₁₀₁	1	1	yes	characteristic
204	C₂²×C₄	—	101	1	no	—
205	C₂²×C₄	—	101	4	no	—
206	C₂²×C₄	—	101	2	no	—
207	C₂²×C₄	—	101	1	no	—
208	C₂²×C₄	—	101	2	no	—
209	C₂²×C₄	—	202	2	no	—
210	C₂²×C₄	—	202	4	no	—
211	C₂²×C₄	—	202	4	no	—
212	C₂²×C₄	—	202	4	no	—
213	C₂²×C₄	—	202	4	no	—
214	C₂²×C₄	—	202	4	no	—
215	C₂²×C₄	—	202	4	no	—
216	C₂²×C₄	—	202	4	no	—
217	C₂²×C₄	—	202	4	no	—
218	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
219	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
220	C₂²×C₄	C₂²×D₁₀₁	1	2	yes	—
221	C₂²×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
222	C₄⋊C₄	—	2	2	no	—
223	C₄⋊C₄	—	202	4	no	—
224	C₄⋊C₄	—	202	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	2	no	—
227	C₂²⋊C₄	—	202	2	no	—
228	C₂²⋊C₄	—	202	4	no	—
229	C₂²⋊C₄	—	202	4	no	—
230	C₂²⋊C₄	—	202	2	no	—
231	C₂²⋊C₄	—	202	4	no	—
232	C₂²⋊C₄	—	202	2	no	—
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	2	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	101	2	no	—
238	C₂³	—	101	2	no	—
239	C₂³	—	101	2	no	—
240	C₂³	—	101	4	no	—
241	C₂³	—	101	4	no	—
242	C₂³	—	101	2	no	—
243	C₂³	—	404	4	no	—
244	C₂³	—	404	4	no	—
245	C₂³	—	404	8	no	—
246	C₂³	—	202	8	no	—
247	C₂³	—	202	16	no	—
248	C₂³	—	202	1	no	—
249	C₂³	—	202	2	no	—
250	C₂³	—	202	2	no	—
251	C₂³	—	202	1	no	—
252	C₂³	D₄×D₁₀₁	1	2	yes	—
253	C₂³	C₂³×D₁₀₁	1	1	yes	characteristic
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	1	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	101	2	no	—
264	C₂×C₄	—	404	8	no	—
265	C₂×C₄	—	404	8	no	—
266	C₂×C₄	—	404	4	no	—
267	C₂×C₄	—	202	8	no	—
268	C₂×C₄	—	202	2	no	—
269	C₂×C₄	—	202	2	no	—
270	C₂×C₄	—	202	1	no	—
271	C₂×C₄	—	202	2	no	—
272	C₂×C₄	—	202	2	no	—
273	C₂×C₄	—	202	2	no	—
274	C₂×C₄	—	202	2	no	—
275	C₂×C₄	—	202	2	no	—
276	C₂×C₄	—	202	2	no	—
277	C₂×C₄	D₄×D₁₀₁	1	2	yes	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	2	no	—
283	C₂²	—	101	4	no	—
284	C₂²	—	101	8	no	—
285	C₂²	—	101	4	no	—
286	C₂²	—	101	4	no	—
287	C₂²	—	101	8	no	—
288	C₂²	—	404	16	no	—
289	C₂²	—	404	1	no	—
290	C₂²	—	404	1	no	—
291	C₂²	—	404	2	no	—
292	C₂²	—	202	2	no	—
293	C₂²	—	202	4	no	—
294	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
295	C₂²	C₂×D₄×D₁₀₁	1	1	yes	characteristic
296	C₂²	#41×D₁₀₁	1	1	yes	characteristic
297	C₂²	#41×D₁₀₁	1	2	yes	—
298	C₂²	#41×D₁₀₁	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	2	no	—
302	C₄	—	4	1	no	—
303	C₄	—	404	2	no	—
304	C₄	—	404	2	no	—
305	C₄	—	404	1	no	—
306	C₄	—	202	2	no	—
307	C₂	—	4	2	no	—
308	C₂	—	101	8	no	—
309	C₂	—	404	2	no	—
310	C₂	#121×D₁₀₁	1	1	yes	characteristic
311	C₂	#123×D₁₀₁	1	1	yes	characteristic
312	C₂	#123×D₁₀₁	1	2	yes	—
313	C₂	#124×D₁₀₁	1	1	yes	characteristic
314	C₂	#126×D₁₀₁	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₀₁
commutator, socle, cosocle	C₂³×C₁₀₁	C₂⁴
Fitting	#397×C₁₀₁	C₂
prime radical	C₂⁴×D₁₀₁	C₂²
hypercenter	#397	D₁₀₁
hypocenter	C₁₀₁	C₂×#397
LCS	G ▷ C₂³×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ #397

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₂×C₄⋊C₄×D₁₀₁	C₂	semidirect
4. #20265106	C₂	semidirect
5. #20265105	C₂	semidirect
6. #20265104	C₂	semidirect
7. #20265103	C₂	semidirect
8. #20265102	C₂	semidirect
9. #95×D₁₀₁	C₂	semidirect
10. #95×D₁₀₁	C₂	semidirect
11. #397×C₁₀₁	C₂	semidirect
17. C₂×C₄⋊Dic₁₀₁	C₂²	semidirect
18. C₂×#13108053	C₂²	semidirect
28. #15692162	C₂²	semidirect
29. #15692162	C₂²	semidirect
30. #15692162	C₂²	semidirect
31. #15692162	C₂²	semidirect
32. #15692162	C₂²	semidirect
33. C₂×C₄⋊C₄×C₁₀₁	C₂²	semidirect
36. #95×C₁₀₁	C₂²	semidirect
37. #95×C₁₀₁	C₂²	semidirect
156. D₁₀₁	#397	direct (N × H)
158. C₁₀₁	C₂×#397	semidirect
167. #397	D₁₀₁	direct (N × H)
178. C₂×C₄⋊C₄	C₂×D₁₀₁	semidirect
194. #95	C₂×D₁₀₁	semidirect
195. #95	C₂×D₁₀₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₄⋊C₄×D₁₀₁	2
C₂×C₂²⋊C₄×D₁₀₁	2
#20265106	2
#20265105	2
#20265104	2
#20265103	2
#20265102	2
#95×D₁₀₁	2
#397×C₁₀₁	2
C₂×#397	101

Subgroups — #397×D101 #27174355

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #397×D₁₀₁ #27174355